Geoscience Reference
In-Depth Information
TABLE 6.2
Example of a Time Evolution in a 1D Channel Computed Using Explicit Central
Differences,
C
r
= 2, and No Diffusion
Grid Point Number
Total
Amount
+
+
+
Time Step
i
- 3
i
- 2
i
- 1
i
i
1
i
2
i
3
0
0
0
0
1
0
0
0
1
1
0
0.00
-1.00
1.00
1.00
0.00
0
1
2
0
1.00
-2.00
-1.00
2.00
1.00
0
1
3
0
3.00
0.00
-5.00
0.00
3.00
0
1
4
0
3.00
8.00
-5.00
-8.00
3.00
0
1
5
0
-5.00
16.00
11.00
-16.00
-5.00
0
1
6
0
-21.00
0.00
43.00
0.00
-21.00
0
1
7
0
-21.00
-64.00
43.00
64.00
-21.00
0
1
8
0
43.00
-128.00
-85.00
128.00
43.00
0
1
9
0
171.00
0.00
-341.00
0.00
171.00
0
1
10
0
171.00
512.00
-341.00
-512.00
171.00
0
1
11
0
-341.00
1024.00
683.00
-1024.0
-341.00
0
1
the errors have increased. The error growth rate has been higher at a higher Courant
number. To understand the reasons for such instability, we can use the following
principle:
“The influence of a point on its neighbors through advection or diffusion cannot be
negative
.
”
This means that the consequence of increasing the concentration in one point
can never be a reduction in any of its neighboring points. In order to guarantee
the respect of this principle, no coefficient of the grid point values in Equation
(6.14) can be negative. If a coefficient is null, there is no influence. In Equation
(6.14), the coefficient of
C
i
+1
is negative whatever the Courant number. As a
consequence, the higher the concentration in that point, the smaller the concen-
tration in point
i
.
This method can be stabilized by adding diffusion. For example, if diffusion is
considered, Equation (6.14) becomes
t
t
t
+
∆
t
t
CC
t
−
U
CC
x
−
CCC
x
−+
2
i
i
=
i
−
1
i
+
1
+
ν
i
−
1
i
i
+
1
2
∆
∆
∆
Ut
x
∆
∆
ν
∆
∆
t
ν
∆
t
Ut
x
∆
∆
ν
∆
∆
t
1
2
1
2
t
+
∆
t
t
t
C
i
t
C
=
+
C
+−
12
C
+−
+
i
2
i
−
1
2
i
2
+
1
x
∆
x
x
(6.15)
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